The scalar invariants of a general gravitational metric

Abstract

A set of fourteen independent scalar invariants of the second order, associated with a general four-dimensional Riemannian metric, is obtained. The set is found to reduce only to four independent invariants in the case of a general spherically symmetrical line-element. Moreover it is shown that the necessary and sufficient conditions that a general spherically symmetrical line-element may represent (i) a flat space, (ii) a Riemannian space of constant curvature, (iii) a Riemannian space conformal to a flat space, (iv) a Riemannian space of class one, can be expressed in terms of these invariants. A new line-element which is the first of its kind known to us is also given, representing the gravitational field due to radiation flowing in one direction and for which all the scalar invariants of the second order vanish.